Place chicken, with giblets, in kettle. Add chicken broth to cover and salt to taste. Bring to boil and simmer until chicken is tender without being dry, 35 to 45 minutes. Let cool. Remove chicken from broth and take meat from bones. Shred meat, cover, and set aside. Return skin and bones to kettle and cook broth down 30 minutes or longer. There should be 4 to 6 cups broth. Strain and reserve broth. Discard the skin and bones.
Meanwhile, put tomatoes in saucepan and cook down to half original volume, stirring. There should be 1 1/2 cups.
Melt 3 tablespoon butter in saucepan, add flour, stirring to blend with wire whisk. When blended and smooth, add 1 cup of reserve hot broth and the cream, stirring rapidly with the whisk. When thickened and smooth, add nutmeg, salt and pepper to taste. Continue cooking, stirring occasionally, about 10 minutes. Set aside.
If the mushrooms are small or button, leave them whole. Otherwise, cut them in half or quarter them. Heat 1 tablespoon butter in small skillet and add mushrooms. Cook, shaking the skillet occasionally, and stirring mushrooms until golden brown. Set aside. Heat the remaining 3 tablespoons butter in deep skillet and add onions. Cook, stirring, until wilted. Add celery and bell peppers. Cook, stirring, about 5 minutes. Do not overcook. Vegetables should remain crisp-tender. Add garlic, beef and pork. Cook, stirring, breaking up the meat. Cook just until the meat loses it red color. Add bay leaf. Add tomatoes and the white sauce made with chicken broth. Add the mushrooms.
Cook spaghettini in boiling salted water until just tender. Do not overcook. Remember that it will cook again when blended with chicken and meat sauce. Drain spaghettini and run under cold running water.
Spoon enough of the meat sauce over the bottom of a 5- to 6-quart casserole to cover it lightly. Add about 1/3 of the spaghettini. Add about 1/3 of the shredded chicken, a layer of meat sauce, a layer of grated Cheddar cheese and another layer of spaghettini. Continue making layers, ending with a layer of spaghettini topped with a thin layer of meat sauce and grated Cheddar cheese. Pour in up to 2 cups reserved chicken broth, or enough to almost but not quite cover the top layer of spaghettini. Cover and let casserole stand 4 hours or longer. If liquid is absorbed as the dish stands, add a little more chicken broth. When dish is baked and served, sauce will be just a little soupy rather than thick and clinging.
Preheat oven to 350 degrees F. Place casserole on top of stove and bring it just to boiling. Cover and place it in the oven. Bake 15 minutes and uncover. Bake 15 minutes longer or until casserole is hot and bubbling throughout, and starting to brown on top. Serve immediately with grated Parmesan on the side.